Is it permissible to pay off an old debt owed to one's mother from Zakat, and is it permissible to pay the Zakat amount in installments and deposit it with a trustworthy person to deliver it to those who are entitled to it?

It is permissible to pay off a mother's debt from Zakat funds if she is unable to repay it, because it is not an obligation upon the child. As for the family's squandering of the money paid from Zakat, they should be advised and warned against extravagance, and urged towards frugality. It is not permissible to pay Zakat in installments after it has become due, but it is permissible before it becomes due.